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Chapter 0: Problem 16

Sketching a Graph by Point Plotting In Exercises \(7-16,\) sketch the graph of the equation by point plotting. \(y=\frac{1}{x+2}\)

Short Answer

Expert verified
The graph of the function \(y=\frac{1}{x+2}\) plots points around a vertical asymptote at \(x=-2\). To the left of this asymptote, the graph trends downwards, while to the right, it goes up.

Step by step solution

01

Understand the Equation

The given equation is \(y=\frac{1}{x+2}\). This is a rational function, which means it will have an undefined value where the denominator equals zero, which is \(x = -2\).
02

Choose Values for x

Choose some values for x to plot the points. It's helpful to include values on either side of the undefined point (-2). In this case, you might choose -3, -2.5, -1, 0, 1.
03

Solve for y

Substitute each x value into the equation to find the corresponding y value. For example, if x=-3, \(y = \frac{1}{-3 + 2} = -1\), if x=-2.5, \(y = \frac{1}{-2.5 + 2} = -2\) and so on. Do this for all chosen x values.
04

Plot Points and Draw the Curve

Plot each point on a graph and draw a smooth curve that fits the points. Keep an eye out for the undefined point at \(x=-2\) where the graph will have a vertical asymptote.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertical Asymptotes
When dealing with rational functions like \( y = \frac{1}{x+2} \), vertical asymptotes are key features on the graph. A vertical asymptote occurs when the denominator of the rational function becomes zero, making the function undefined at that x-value. In our equation, the denominator \( x + 2 \) becomes zero when \( x = -2 \). This is the location of the vertical asymptote.
Visualize the vertical asymptote as an imaginary vertical line at \( x = -2 \). The graph will approach this line but will never touch or cross it.
Keep in mind:
  • Vertical asymptotes indicate where the graph of the function may spike upward or downward to infinity.
  • Always look for values of \( x \) where the denominator equals zero to find potential vertical asymptotes.
  • This asymptotic behavior is crucial for sketching the graph accurately.
Plotting Points
Plotting points helps to visualize the graph of a rational function by marking specific coordinates. Begin by selecting several \( x \)-values to input into your function. It's smart to pick values around the vertical asymptote to understand how the graph behaves near these crucial points.
Let's use \( y = \frac{1}{x+2} \):
  • Choose \( x = -3 \), you get \( y = -1 \).
  • For \( x = -2.5 \), find \( y = -2 \).
  • If \( x = 0 \), then \( y = \frac{1}{2} \).
  • Suppose \( x = 1 \), lastly \( y = \frac{1}{3} \).
Plot these calculated points on your graph paper and observe. You should notice a pattern forming, especially around the vertical asymptote. This point plotting technique provides the backbone for drawing the graph's curve.
Rational Functions
A rational function is a fraction where both the numerator and denominator are polynomials. In our exercise, \( y = \frac{1}{x+2} \), the numerator is \( 1 \), a constant, and the denominator is \( x+2 \) a simple linear polynomial.
Important properties include:
  • Rational functions can exhibit holes (removable discontinuities) where both numerator and denominator are zero.
  • They often have horizontal or slant asymptotes, in addition to vertical ones, affecting the behavior at infinity.
  • Simplifying the rational expression is crucial when the numerator or denominator can be factored further.
In our specific case, there's no opportunity for simplification, since the numerator is not a polynomial of \( x \). However, observing these aspects can help when dealing with more complex rational functions.
Asymptotic Behavior
The term 'asymptotic behavior' describes how the graph of a function behaves as it goes towards an asymptote. This behavior is unmistakable in rational functions due to their denominator-driven characteristics.
Consider these types of asymptotic behaviors:
  • **Near Vertical Asymptotes**: As the value of \( x \) approaches the vertical asymptote (\( x = -2 \) for our function), the function \( y \) will tend toward positive or negative infinity. This means the graph will spike upwards or downwards steeply.
  • **At Infinity**: For many rational functions, there are also horizontal asymptotes that show the value the function approaches as \( x \) goes to either positive or negative infinity. Our function's horizontal asymptote in this case is \( y = 0 \).
  • **Graph Behavior**: Observing how the function stretches towards the asymptotes helps in accurately sketching the curve of the graph.
Understanding asymptotic behavior allows us to predict the direction and the trend of a graph close to undefined regions or far from the origin.

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